Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dcned GIF version

Theorem dcned 2314
 Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.)
Hypothesis
Ref Expression
dcned.eq (𝜑DECID 𝐴 = 𝐵)
Assertion
Ref Expression
dcned (𝜑DECID 𝐴𝐵)

Proof of Theorem dcned
StepHypRef Expression
1 dcned.eq . . 3 (𝜑DECID 𝐴 = 𝐵)
2 dcn 827 . . 3 (DECID 𝐴 = 𝐵DECID ¬ 𝐴 = 𝐵)
31, 2syl 14 . 2 (𝜑DECID ¬ 𝐴 = 𝐵)
4 df-ne 2309 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
54dcbii 825 . 2 (DECID 𝐴𝐵DECID ¬ 𝐴 = 𝐵)
63, 5sylibr 133 1 (𝜑DECID 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 819   = wceq 1331   ≠ wne 2308 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698 This theorem depends on definitions:  df-bi 116  df-dc 820  df-ne 2309 This theorem is referenced by:  nn0n0n1ge2b  9137  algcvgblem  11736
 Copyright terms: Public domain W3C validator